An algebraic spectrum condition

A condition, necessary and sufficient for the existence of a vacuum representation with positive energy of the quasilocal algebra, is formulated.     

An algebraic condition for the approach to equilibrium of an open N-level system

We give an algebraic condition in order that a completely positive dynamical semigroup of an N-level system has a unique (invariant) equilibrium state and that every initial state approaches this equilibrium state as t . We apply our result to a semigroup arising in the weak coupling limit.     

Representations obeying the spectrum condition

We show that every properly infinite, injective von Neumann algebra acting on a separable Hilbert space is isomorphic to the weak closure of some translation covariant representation, obeying the spectrum condition for the generators of the translation group, of theC*-algebra of quasilocal observables of a free massless spinor field. We construct explicitly such representations in the case ofII andIII factors, 0<<1.     

Translation group and spectrum condition

Let {A, ^d ,} be aC*-dynamical system, where ^d is thed-dimensional vector group. LetV be a convex cone in ^d and its dual cone. We will characterize those representations ofA with the properties (i) _a ,a ^d is weakly inner, (ii) the corresponding unitary representationU(a) is continuous, and (iii) the spectrum ofU(a) is contained in .     

On modular inclusion and spectrum condition

For a family of von Neumann algebras, we investigate the relations between spectrum condition, half-sided modular inclusion, and reflection symmetry. It is shown that the geometric interpretation of modular actions puts strong restrictions on the family.     

An algebraic treatment of Franck-Condon-transitions

A purely algebraic evaluation of Franck-Condon-Integrals, of transition rates for energy conserving Franck-Condon-transitions and of the related thermal averaged quantities is performed.     

An algebraic approach to discrete mechanics

Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular automata in which time proceeds in integer steps and the configuration space is discrete. In particular, we derive an analog of the Euler-Lagrange equation from a variational principle, and prove an analog of Noether's theorem. We also construct a symplectic structure on the analog of the phase space, and prove that it is preserved by time evolution.     

An algebraic approach to soliton solutions

Within the scope of a new algebraic method for the construction of soliton solutions of nonlinear equations [10–13], concrete examples of the system are considered which are connected with the algebra sl(2, R).     

An algebraic spin and statistics theorem

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional spacetime. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.Dedicated to Hans Borchers on the occasion of his seventieth birthdaySupported in part by MURST and CNR-GNAFA.     

An algebraic approach to form factors

An associative *-algebra is introduced (containing a TTR-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each T-invariant linear functional over the algebra automatically satisfies all the form factor axioms. It is argued that this answers the question (posed in the functional Bethe ansatz) how to select the dynamically correct representations of the TTR-algebra. Applied to the case of integrable QFTs with diagonal factorized scattering theory a universal formula for the eigenvalues of the conserved charges emerges.     

An algebraic isometric embedding of Kruskal space-time

Recent results concerning isometric embeddings of Schwarzschild space-time and its maximal extension are given. It is shown that Kruskal's manifold can be embedded into R⁷ as an algebraic subvariety. A discussion of the importance of this embedding is given.     

An algebraic analog of the virasoro group

The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it acts as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of 2D topological gravity. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Supported in part by the NSF.     

On the spectrum of internal symmetries in the algebraic quantum field theory

It is shown that the point spectrum of internal symmetries is always symmetric. It is a group provided the intersection of all local subalgebras is trivial.     

An algebraic framework for the supersymmetry theory

In the absence of a rigorous mathematical definition (or proof of existence and consistency) of the “g-dimensional space” whose points are labelled by the four (commutative) Minkowski coordinates x_n and four anticommuting coordinates θ_n, a clearly consistent algebraic scheme is presented in the frame of which the concepts used in the theory of the supersymmetry have a well established mathematical meaning.     

An algebraic approach to the non-symmetric Macdonald polynomial

In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are simultaneous eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them.     

An algebraic multigrid solver for transonic flow problems

This article presents the latest developments of an algebraic multigrid (AMG) based on full potential equation (FPE) solver for transonic flow problems with emphasis on advanced applications. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic flow) to hyperbolic (supersonic flow). The flow solver is capable of dealing with flows from subsonic to transonic and supersonic conditions and is based on structured body-fitted grids approach for treating complex geometries. The computational method was demonstrated on a variety of problems to be capable of predicting the shock formation and achieving residual reduction of roughly an order of magnitude per cycle both for elliptic and hyperbolic problems, through the entire range of flow regimes, independent of the problem size (resolution).     

An algebraic approach to the planar coloring problem

We point out a general relationship between the planar coloring problem withQ colors and the Temperley-Lieb algebra with parameter . This allows us to give a complete algebraic reformulation of the four color result, and to give algebraic interpretations of various other aspects of planar colorings.Work supported in part by NSF Grant #DMS-882602, the program for Mathematics and Molecular Biology, UC Berkeley, and a visiting fellowship of the Japan Society for the promotion of science at Kyoto University, Kyoto, JapanWork supported in part by DOE Contact #DE-AC02-76ERO3075 and by a Packard Fellowship for Science and Engineering     

An algebraic approach to the restoration of images

Two different matrix algorithms are described for the restoration of blurred pictures. These are illustrated by numerical examples.